Derangement formula

Setup. Let $A_i$ = permutations fixing element $i$. We want permutations fixing no element, i.e. $\overline{A_1\cup\cdots\cup A_n}$.

Count. $|A_{i_1}\cap\cdots\cap A_{i_k}|=(n-k)!$ (permutations fixing $k$ elements arbitrarily).

Inclusion-exclusion.
$$D_n=\sum_{k=0}^n(-1)^k\binom{n}{k}(n-k)!=n!\sum_{k=0}^n\frac{(-1)^k}{k!}.$$

Conclusion. $D_n=n!\sum_{k=0}^n\frac{(-1)^k}{k!}$.